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On estimating population characteristics from record‐breaking observations II. Nonparametric results
Author(s) -
Samaniego Francisco J.,
Whitaker Lyn R.
Publication year - 1988
Publication title -
naval research logistics (nrl)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 68
eISSN - 1520-6750
pISSN - 0894-069X
DOI - 10.1002/1520-6750(198804)35:2<221::aid-nav3220350207>3.0.co;2-2
Subject(s) - estimator , nonparametric statistics , quantile , mathematics , consistency (knowledge bases) , asymptotic distribution , population , statistics , parametric statistics , sequence (biology) , econometrics , discrete mathematics , demography , sociology , biology , genetics
Consider an experiment in which only record‐breaking values (e.g., values smaller than all previous ones) are observed. The data available may be represented as X 1 , K 1 , X 2 , K 2 , …, where X 1 , X 2 , … are successive minima and K 1 , K 2 , … are the numbers of trials needed to obtain new records. Such data arise in life testing and stress testing and in industrial quality‐control experiments. When only a single sequence of random records are available, efficient estimation of the underlying distribution F is possible only in a parametric framework (see Samaniego and Whitaker [9]). In the present article we study the problem of estimating certain population quantiles nonparametrically from such data. Furthermore, under the assumption that the process of observing random records can be replicated, we derive and study the nonparametric maximum‐likelihood estimator F̂ of F . We establish the strong uniform consistency of this estimator as the number of replications grows large, and identify its asymptotic distribution theory. The performance of F̂ is compared to that of two possible competing estimators.